Consider two bases $b_1=\begin{pmatrix}1\\ 0\end{pmatrix}$, $b_2=\begin{pmatrix}0\\ 1\end{pmatrix}$, and the other two bases $c_1=\begin{pmatrix}1\\ 0\end{pmatrix}$, $c_2=\begin{pmatrix}2\\ 1\end{pmatrix}$, suppose $x=\begin{pmatrix}1\\ 1\end{pmatrix}$, so $[x]_B=\begin{pmatrix}1\\ 1\end{pmatrix}$, now I need to know what $[x]_C$ is, and I got a confusing answer, below is what I have done
the change of coordinate matrix is $\begin{pmatrix}[b_1]_C & [b_2]_C\end{pmatrix}=\begin{pmatrix} 1& -2\\0 & 1\end{pmatrix}$, since $b_1=1*c_1+0*c_2$ and $b_2=-2*c_1+1*c_2$
so $[x]_C=\begin{pmatrix} 1& -2\\0 & 1\end{pmatrix}\begin{pmatrix}1\\ 1\end{pmatrix}=\begin{pmatrix}-1\\ 1\end{pmatrix}$, but this is obviously incorrect, where did I go wrong?
Why is it incorrect? $$[x]_C = \begin{pmatrix}-1\\ 1\end{pmatrix}$$ means: $$x = -1c_1+1c_2 = -\begin{pmatrix}1\\ 0\end{pmatrix}+\begin{pmatrix}2\\ 1\end{pmatrix} = \begin{pmatrix}-1+2\\ 1\end{pmatrix}= \begin{pmatrix}1\\ 1\end{pmatrix}$$ and that's the $x$ you started with, right?